Toric polar maps and characteristic classes

TL;DR

This paper establishes a deep connection between the multidegrees of toric polar maps of hypersurfaces and the Chern-Schwartz-MacPherson classes of specific open sets, providing new formulas and constructions in algebraic geometry.

Contribution

It proves the equivalence of multidegrees of toric polar maps with Chern-Schwartz-MacPherson class coefficients and constructs hypersurfaces with birational toric polar maps.

Findings

Multidegrees of toric polar maps match Chern-Schwartz-MacPherson class coefficients.

Degree of the toric polar map equals the signed topological Euler characteristic.

Explicit formulas for plane curves and examples of hypersurfaces with birational toric polar maps.

Abstract

Given a hypersurface in a complex projective space, we prove that the multidegrees of its toric polar map agree, up to sign, with the coefficients of the Chern-Schwartz-MacPherson class of a distinguished open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes. In particular, the degree of the toric polar map is given by the signed topological Euler characteristic of the distinguished open set. For plane curves, a precise formula for the degree of the toric polar map is obtained in terms of local invariants. Finally, we construct families, in arbitrary dimension, of irreducible hypersurfaces whose toric polar map is birational.